scholarly journals Existence of solutions for a boundary problem involving p(x)-biharmonic operator

2012 ◽  
Vol 31 (1) ◽  
pp. 179 ◽  
Author(s):  
Abdel Rachid El Amrouss ◽  
Anass Ourraoui
Keyword(s):  
Boundary Problem ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Biharmonic Operator ◽  
Technical Approach ◽  
Three Solutions

In this paper, we establish the existence of at least three solutions to a boundary problem involving the p(x)-biharmonic operator. Our technical approach is based on theorem obtained by B. Ricceri's variational principale and local mountain pass theorem without (Palais.Smale) condition.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

scholarly journals GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

2020 ◽  
Vol 6 (1) ◽  
pp. 30
Author(s):  
Hassan Belaouidel ◽  
Anass Ourraoui ◽  
Najib Tsouli
Keyword(s):  
Mountain Pass Theorem ◽  
Type Equation ◽  
Robin Boundary Condition ◽  
Mountain Pass ◽  
Technical Approach ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Quasilinear Problems ◽  
Robin Boundary ◽  
Laplace Type

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$\left\{\begin{array}{lll}-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.\end{array}\right.$$ Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.

scholarly journals Existence of solutions for a class of strongly coupled $p(x)-$ laplacian system

2018 ◽  
Vol 36 (4) ◽  
pp. 183-195
Author(s):  
Eada Ahmed Al Zahrani ◽  
Mohamed Ali Mourou ◽  
Kamel Saoudi
Keyword(s):  
Weak Solution ◽  
Nonlinear System ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Main Argument ◽  
Strongly Coupled ◽  
Laplacian Operators

We prove the existence of a non trivial weak solution for certain class of strongly coupled nonlinear system containing the ($p(x)-q(x))$ laplacian operators using as main argument the mountain pass Theorem of {\sc Ambrosetti-Rabinowitz}.

scholarly journals On a Class of PDE Involving p-Biharmonic Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Abdelouahed El Khalil
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourth Order ◽  
Mountain Pass Theorem ◽  
Nonlinear Partial Differential Equation ◽  
Existence Of Solution ◽  
Mountain Pass ◽  
Biharmonic Operator ◽  
Partial Differential

The existence of solution for a fourth-order nonlinear partial differential equation (PDE) class involving p-biharmonic operator Δ(|Δu|p−2Δu)=λρ(x)|u|q−2u  in  Ω,  u=Δu=0,  on⁡  ∂Ω, is proved by applying mountain pass theorem and a local minimization.

scholarly journals Existence of solutions for a class of $p(x)$-curl systems arising in electromagnetism without (A-R) type conditions

2020 ◽  
Vol 51 (3) ◽  
pp. 187-200
Author(s):  
Nguyen Thanh Chung
Keyword(s):  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

In this paper, we study the existence and multiplicity of solutions for a class of of $p(x)$-curl systems arising in electromagnetism.  Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the $p(x)$-curl operator in [4, 15].

Existence results for a Kirchhoff type equation in Orlicz–Sobolev spaces

2017 ◽  
Vol 8 (3) ◽  
Author(s):  
EL Miloud Hssini ◽  
Najib Tsouli ◽  
Mustapha Haddaoui
Keyword(s):  
Variational Principle ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Existence Results ◽  
Mountain Pass ◽  
Nonlocal Problems ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation

AbstractIn this paper, based on the mountain pass theorem and Ekeland’s variational principle, we show the existence of solutions for a class of non-homogeneous and nonlocal problems in Orlicz–Sobolev spaces.

scholarly journals Multiple solutions for a problem with resonance involving thep-Laplacian

1998 ◽  
Vol 3 (1-2) ◽  
pp. 191-201 ◽  
Author(s):  
C. O. Alves ◽  
P. C. Carrião ◽  
O. H. Miyagaki
Keyword(s):  
Variational Principle ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Mountain Pass Theorem ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Three Solutions ◽  
Bounded Functions

In this paper we will investigate the existence of multiple solutions for the problem(P)                                                         −Δpu+g(x,u)=λ1h(x)|u|p−2u,     in     Ω,    u∈H01,p(Ω)whereΔpu=div(|∇u|p−2∇u)is thep-Laplacian operator,Ω⫅ℝNis a bounded domain with smooth boundary,handgare bounded functions,N≥1and1<p<∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).

Equations of p-Laplacian Type in Unbounded Domains

2002 ◽  
Vol 2 (3) ◽  
Author(s):  
Pablo L. De Nápoli ◽  
M. Cristina Mariani
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Unbounded Domains ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Solutions To Equations

AbstractThis work is devoted to study the existence of solutions to equations of the p Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.

scholarly journals Existence of solutions for an elliptic p(x)-Kirchhoff-type systems in unbounded domain

2018 ◽  
Vol 36 (3) ◽  
pp. 193-205 ◽  
Author(s):  
Abdelamlek Brahim ◽  
Ali Djellit ◽  
Saadia Tas
Keyword(s):  
Variational Method ◽  
Elliptic System ◽  
Unbounded Domain ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Main Tool ◽  
Type Systems ◽  
Mountain Pass ◽  
Nonlocal Term ◽  
Kirchhoff Type

In this paper we study of the existence of solutions for a class of elliptic system with nonlocal term in R^{N}. The main tool used is the variational method, more precisely, the Mountain Pass Theorem.

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